The aim of this article is to study the well-posedness and properties of a fast-slow system which is related with brain lactate kinetics. In particular, we prove the existence and uniqueness of nonnegative solutions and obtain linear stability results. We also give numerical simulations with different values of the small parameter $\varepsilon$ and compare them with experimental data.

Figure 1.
Schematic representation of lactate exchanges in a local brain part. There is a cotransport through the brain-blood barrier, a blood flow, cell creation and consumption and interactions between a cell and its neighborhood. Interactions are described in the main text

Figure 3.
Intracellular and capillary lactate dynamics with nonconstant functions $J$ and $F$. On the left, the intracellular lactate trajectory is upper bounded. On the right, the capillary lactate trajectory is upper bounded too, but has an initial dip. At the bottom the orbit is typical of fast-slow systems

Figure 4.
Intracellular and capillary lactate dynamics with constant functions $J$ and $F$. The intracellular lactate trajectory (on the left) and the capillary lactate trajectory (on the right) are both upper bounded and reach the corresponding steady state. The trajectories for the original system (with $\varepsilon >0$) are also lower bounded. On the right top corner the capillary lactate of the original system has an initial dip, while it does not exist on the capillary lactate curve of the limit system (right bottom corner)

Figure 5.
Dynamics for different values of $\varepsilon$. On the left intracellular; the lactate trajectories seem not to differ a lot. On the right, the value of $\varepsilon$ is related to the dip stiffness for the capillary lactate trajectories

Figure 6.
Dynamics for different values for $J$. On the right, the intracellular lactate trajectories are divided into two groups : for $J \in \{1, 0.1, 0.01 \}$, the concentration seems to explode, while for $J \in \{0.001, 0.0001\}$, it seems more stable. On the right the capillary lactate trajectories are devided into these two groups. For the first one, we can see a dip, while, for the second one, the steady state is not quickly reached

Figure 7.
Lactate concentration changes in a local brain part. Lactate concentration is given in mM (vertical axis) and time in days (horizontal axis). The red dots stand for medical data values, while the model simulations are displayed in continuous lines. While the four first patients exhibit Grompertz growth of their brain lactate concentration, patient 5 lactate concentration decreases in time. All the dynamics simulations tend to the steady state given in section 2

Isao Kato.
Well-posedness
for the Cauchy problem of
the Klein-Gordon-Zakharov system
in four and more spatial dimensions.
Communications on Pure & Applied Analysis,
2016, 15
(6)
: 2247-2280.
doi: 10.3934/cpaa.2016036